12
PAUL E. EHRLICH AND KEVIN L. EASLEY
Ka Kb K[cRd]ab[eKf]
is not zero. We shall say that a space-time
satisfying this condition satisfies the generic condition."
So from the cynical viewpoint, one might take the following interpretation from
this paragraph, which the first author did for many years in perfect contentment:
(4.6) generic condition
?
physically realistic.
On the other hand, "generic" has a precise meaning in differential geometry and
topology; a condition is said to hold generically when it holds on an open, dense
subset of the space in question. It never occurred to the first author to ponder
how ( 4.6) interfaced with this more precise definition of "generic," so he was thus
delighted in the early 1990s to receive two preprints from
J.
Beem and S. Harris,
published as [27] and [28], the first with the especially charming title "The generic
condition is generic." Since the most precise results in these papers are a bit com-
plicated to state, we will content ourselves here with just giving four of the more
easily stated results obtained in these two publications:
1. Ric( w, w)
#
0 implies w is generic.
2. All vectors in TpM are nongeneric implies that the curvature tensor at p
vanishes identically.
3. Constant curvature implies all null vectors are nongeneric.
4. If ( M
4
,
g)
does not have constant sectional curvature at
p,
then the generic
null directions at p form an open dense subset of the two-sphere of all null
directions at
p.
Thus, one could interpret ( 4) as stating that "the generic condition is generic
after all," since a physically realistic universe should probably not have constant
sectional curvature.
5. The Stability of Geodesic Completeness Revisited
In the First Edition of Global Lorentzian Geometry
[21],
a short Section
6.1 was written, entitled "Stable Properties of Lor(M) and Con(M)," which was
partly inspired by results of Lerner [72]. A motivation for this type of investiga-
tion in General Relativity had been provided by the hypotheses in the Singularity
Theorems. If a condition held on an open subset of metrics in the space Lor(M)
of all Lorentzian metrics for a given smooth manifold M, then philosophically a
robuster theorem would result since this part of the hypotheses would remain true
under suitable perturbations of the given metric, desirable since measurements can
not be made with infinite precision.
A result quoted in this Section 6.1 was the
cr
-stability of geodesic complete-
ness in Lor(M) for all r
2:
2. In the remaining two sections of Chapter 6, based
on Beem and Ehrlich [22], a question raised in [72] was studied-the stability of
timelike and null geodesic incompleteness for Robertson-Walker space-times. Here,
a Robertson-Walker space-time was taken to be a warped product
M =(a, b) x
1
H
where (H, h) was a homogeneous Riemannian manifold and the metric tensor had
the form
(5.1) g=-dt
2
+fh.
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